In [1]}, Mazur and Tate proposed several conjectures refining the BSD conjectures equivariantly.
Since the initial Fitting ideals are used in the formulation of their weak main conjecture, it would be regarded as the first approach towards the structure of Selmer groups.
Theorem REF  refines their weak vanishing conjecture [1]} by making their inequality into the equality when the character is trivial and also refines their weak main conjecture [1]} and a “Birch–Swinnerton-Dyer type" conjecture [1]} by considering all the higher Fitting ideals but with no equivariant variation.
We refer to [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]} for the recent developments.
